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97aa5af | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 | """ 3-d rigid body transfomation group and corresponding Lie algebra. """
import torch
from .sinc import sinc1, sinc2, sinc3
from . import so3
def twist_prod(x, y):
x_ = x.view(-1, 6)
y_ = y.view(-1, 6)
xw, xv = x_[:, 0:3], x_[:, 3:6]
yw, yv = y_[:, 0:3], y_[:, 3:6]
zw = so3.cross_prod(xw, yw)
zv = so3.cross_prod(xw, yv) + so3.cross_prod(xv, yw)
z = torch.cat((zw, zv), dim=1)
return z.view_as(x)
def liebracket(x, y):
return twist_prod(x, y)
def mat(x):
# size: [*, 6] -> [*, 4, 4]
x_ = x.view(-1, 6)
w1, w2, w3 = x_[:, 0], x_[:, 1], x_[:, 2]
v1, v2, v3 = x_[:, 3], x_[:, 4], x_[:, 5]
O = torch.zeros_like(w1)
X = torch.stack((
torch.stack(( O, -w3, w2, v1), dim=1),
torch.stack(( w3, O, -w1, v2), dim=1),
torch.stack((-w2, w1, O, v3), dim=1),
torch.stack(( O, O, O, O), dim=1)), dim=1)
return X.view(*(x.size()[0:-1]), 4, 4)
def vec(X):
X_ = X.view(-1, 4, 4)
w1, w2, w3 = X_[:, 2, 1], X_[:, 0, 2], X_[:, 1, 0]
v1, v2, v3 = X_[:, 0, 3], X_[:, 1, 3], X_[:, 2, 3]
x = torch.stack((w1, w2, w3, v1, v2, v3), dim=1)
return x.view(*X.size()[0:-2], 6)
def genvec():
return torch.eye(6)
def genmat():
return mat(genvec())
def exp(x):
x_ = x.view(-1, 6)
w, v = x_[:, 0:3], x_[:, 3:6]
t = w.norm(p=2, dim=1).view(-1, 1, 1)
W = so3.mat(w)
S = W.bmm(W)
I = torch.eye(3).to(w)
# Rodrigues' rotation formula.
#R = cos(t)*eye(3) + sinc1(t)*W + sinc2(t)*(w*w');
# = eye(3) + sinc1(t)*W + sinc2(t)*S
R = I + sinc1(t)*W + sinc2(t)*S
#V = sinc1(t)*eye(3) + sinc2(t)*W + sinc3(t)*(w*w')
# = eye(3) + sinc2(t)*W + sinc3(t)*S
V = I + sinc2(t)*W + sinc3(t)*S
p = V.bmm(v.contiguous().view(-1, 3, 1))
z = torch.Tensor([0, 0, 0, 1]).view(1, 1, 4).repeat(x_.size(0), 1, 1).to(x)
Rp = torch.cat((R, p), dim=2)
g = torch.cat((Rp, z), dim=1)
return g.view(*(x.size()[0:-1]), 4, 4)
def inverse(g):
g_ = g.view(-1, 4, 4)
R = g_[:, 0:3, 0:3]
p = g_[:, 0:3, 3]
Q = R.transpose(1, 2)
q = -Q.matmul(p.unsqueeze(-1))
z = torch.Tensor([0, 0, 0, 1]).view(1, 1, 4).repeat(g_.size(0), 1, 1).to(g)
Qq = torch.cat((Q, q), dim=2)
ig = torch.cat((Qq, z), dim=1)
return ig.view(*(g.size()[0:-2]), 4, 4)
def log(g):
g_ = g.view(-1, 4, 4)
R = g_[:, 0:3, 0:3]
p = g_[:, 0:3, 3]
w = so3.log(R)
H = so3.inv_vecs_Xg_ig(w)
v = H.bmm(p.contiguous().view(-1, 3, 1)).view(-1, 3)
x = torch.cat((w, v), dim=1)
return x.view(*(g.size()[0:-2]), 6)
def transform(g, a):
# g : SE(3), * x 4 x 4
# a : R^3, * x 3[x N]
g_ = g.view(-1, 4, 4)
R = g_[:, 0:3, 0:3].contiguous().view(*(g.size()[0:-2]), 3, 3)
p = g_[:, 0:3, 3].contiguous().view(*(g.size()[0:-2]), 3)
if len(g.size()) == len(a.size()):
b = R.matmul(a) + p.unsqueeze(-1)
else:
b = R.matmul(a.unsqueeze(-1)).squeeze(-1) + p
return b
def group_prod(g, h):
# g, h : SE(3)
g1 = g.matmul(h)
return g1
class ExpMap(torch.autograd.Function):
""" Exp: se(3) -> SE(3)
"""
@staticmethod
def forward(ctx, x):
""" Exp: R^6 -> M(4),
size: [B, 6] -> [B, 4, 4],
or [B, 1, 6] -> [B, 1, 4, 4]
"""
ctx.save_for_backward(x)
g = exp(x)
return g
@staticmethod
def backward(ctx, grad_output):
x, = ctx.saved_tensors
g = exp(x)
gen_k = genmat().to(x)
# Let z = f(g) = f(exp(x))
# dz = df/dgij * dgij/dxk * dxk
# = df/dgij * (d/dxk)[exp(x)]_ij * dxk
# = df/dgij * [gen_k*g]_ij * dxk
dg = gen_k.matmul(g.view(-1, 1, 4, 4))
# (k, i, j)
dg = dg.to(grad_output)
go = grad_output.contiguous().view(-1, 1, 4, 4)
dd = go * dg
grad_input = dd.sum(-1).sum(-1)
return grad_input
Exp = ExpMap.apply
#EOF
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